Question: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{9x^3 - 81x^2 + 180x}{7x^3 - 175x}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ r = \dfrac {9x(x^2 - 9x + 20)} {7x(x^2 - 25)} $ $ r = \dfrac{9x}{7x} \cdot \dfrac{x^2 - 9x + 20}{x^2 - 25} $ Simplify: $ r = \dfrac{9}{7} \cdot \dfrac{x^2 - 9x + 20}{x^2 - 25}$ Since we are dividing by $x$ , we must remember that $x \neq 0$ Next factor the numerator and denominator. $ r = \dfrac{9}{7} \cdot \dfrac{(x - 5)(x - 4)}{(x - 5)(x + 5)}$ Assuming $x \neq 5$ , we can cancel the $x - 5$ $ r = \dfrac{9}{7} \cdot \dfrac{x - 4}{x + 5}$ Therefore: $ r = \dfrac{ 9(x - 4)}{ 7(x + 5)}$, $x \neq 5$, $x \neq 0$